Understanding the Birthday Paradox

[…] After reading  another math explanation on why that’s true, I know that I understand it now. Sure, I might not be able to repeat (or fully understand) the math equations which generate the percentage, but I can identify the bottom line of understanding — when written in POE (plain ol’ English): […]

Heyy ;; i have no clue how to do this!

I think that the math behind this birthday paradox is wrong…
The chance of two people having same birthdays is 1/365 = 0.0027397

therefore p(n)= 0.0027397 ^C(n,2)
if we take an example of 23 people
we get p(23)= 0.0027397 ^ 253 ~=0
so how is it possible??

Hi, you’re correct 1/365 is the chance of 2 people having the same birthday. However, (1/365)^253 would be the chance of 253 people having the same birthday! (Which, as you see, is pretty close to zero).

For this problem, it’s important not to mix up 1/365 (the chance of 1 collision) and 364/365 (the chance of no collision). We first find the chance that somehow, everyone manages to be different:

p(23 people have different birthdays) = (364/365)^253

If there is a 40% chance that everyone is different, there is 1-40% = 60% chance that there was an overlap somewhere. Hope this helps. (Technically, we are assuming independent events but that subtlety is not important for the main point).

hi,
(364/365)^253 means that 253 people have different birthdays

when you check this for 366 people , there is a >=100% chance for the birthday paradox.
but when you use this fomula we get the answer as 1 - 2.6 * 10^-80 which is less than 1

why is it so??

AND I have never seen two people having the same birthday in my group which has a greater strength than 23.this cannot be a coincidence!!!

I still doubt that there is a 50% chance of people having the same birthday

Hi, when you make the probability like (364/365)^253, you are assuming independent events. What this means is that each comparison is “fresh”, with no memory of the past. It would be like having 2 people pick the same number out of 365, and choosing a different number each time.

This approximation makes the math easier, and is ok for small values. If you want the actual %, take a look at Appendix A.

Yep, the paradox seems strange, doesn’t it? Take a look at this page and run some experiments on your own to see:

http://betterexplained.com/examples/birthday/birthday.html

As you click “run trial”, you will see the actual match percentage for 23 people approach 50%, which is the predicted one. Hope this helps.

[…] A GUID, or large ID number used in programming, is at no risk of running out. How many are there? Well, we could give everyone a copy of the internet, every second, for a billion years… and still have enough GUIDs to identify each page. See how much bigger that is than “2^128″? (For the geeks: yes, the birthday paradox makes the chance of collision much higher). […]

the math for the birthday paradox is in fact quite simple, the “problem scenario” probability is in fact

364/365 times 363/365 times … times (364-22)/365

you should think like this.

-person one chose a day of the year as a birthday
-person two chose a day of the year as its birthday, BUT DIFFERENT than person one’s choice.
-so on
-person 23 does the same BUT DIFFERENT than previous people choices.

This is exactly what I wrote above in probabilities

Oh yeah,… sorry the last fraction is (365-22)/365

bye

Yep, that’s right. Sometimes that multiplication can be long to do out – see Appendix A for a shortcut.

Thanks for this…im gonna use this as an idea for science fair!

Testing to see if the Birthday Paradox holds true.
23 in a room, 50% chance two will match!

Can’t wait!

Sounds great Brittany! And if you have 75 people at your fair, you’re almost guaranteed to have a match :).

It’s funny. There are actually two birthday paradoxes. The other comes from logic and is actually, actually, according to Quine, a veridical paradox, where it appears to be paradoxical, yet is proven true anyway, the fact that someone turns 7 when they are twenty-eight years old (born feb. 29), much like this birthday paradox.

What is interesting is that the two overlap. So to properly treat the birthday paradox (your version) you would have to take this into account.

So a very interesting treatment would be: what happens to the probability of sharing a birthday when you take into account feb 29, twins, triplets, etc, the fact (i believe) that there are higher frequencies of babies born during certain times of the year than others.

I might work this out, if asked, but I don’t think it would work out to 50% out of 23. It would be interesting to see how close it was though.

[…] Regarding your birthday, whether you are savvy with Hamming’s error correcting code or not, listen to Kalid Azad when he presents Understanding the Birthday Paradox posted at BetterExplained in which he explains the Birthday Paradox from statistics. […]

thx 4 the info it was confusing but really good, im going 2 use this 4 my science fair project

[…] In class I used the example of "The Birthday Paradox" whilst discussing behavioural finance, and was reminded that I also raised the issue - without fully explaining it - during the Quantitative Methods course. To compensate, here's a list of explanations. The paradox comes from the unintuitive finding that the probability that any of 23 random people sharing a birthday is a whopping 50%. The problem is that when I select 23 people in class, there's only a 50-50 chance it comes off. In future I might select 30 students, since then there;s a 70% chance of a match, and I assume this would still be sufficiently surprising to demonstrate the point. The explanation that makes most sense to me is as follows: […]

[…] Better Explained is a website where this guy, Kalid Azad, explains things really well. Like, really well. So far, the three main catagories of explanations are “Math and Numbers,” “Programming and Web Development,” and “Business, Writing & Communication.” But, for example, check out his explanation of one of the more fascinating things that I learned back in Game Theory at CTY (oh good times…): the Birthday Paradox. Yes, if you’re in a room with 22 other people, the likelyhood of two people having the same birthday is just over 50%. Crazy, huh. […]

[…] The Birthday Paradox The Birthday Paradox explained and a interactive example included on the page. Explore posts in the same categories: Paradox […]

Does the dependency matter really at all?? I have just read it once, so maybe I don’t get it yet, but it seems you are just looking for at least 1 match?
50/50 chance of at least one match? If that is the case why would the dependency matter?

It seems since you are looking at each individual group at a time, that each event would be independent from the rest. Therefore looking at each group separately each group has a 1/365th possibility of matching?

hmm
I don’t know